The news was announced live on the station this morning (November 9).
Yates currently presents The Official Chart With Reggie Yates on Sundays, while Kay hosts a Saturday morning show.
Yates will be replaced by Jameela Jamil, who becomes the first ever solo woman to present the Chart Show.
Matt Edmondson replaces Kay in the 10am-1pm Saturday slot. Edmondson also takes over Sara Cox's Sunday show when she moves to Radio 1 to cover Fearne Cotton, who goes on maternity leave at the end of the year.
"I've had an incredible nine-and-a-half years at Radio 1 that has exceeded all of my expectations, it has been a fabulous experience and I've loved the weekends I have spent with my listeners," said Kay.
"However, when I was offered a contract extension I thought long and hard, but after 12 years of working weekends with CBBC, T4 and Radio 1, I felt it was the right time to leave as the girls are now both in school so weekends are more precious than ever."
Yates added: "I have absolutely loved my time hosting the Chart Show. It's been a privilege to deliver to the nation the weekly No.1 song, but after 10 years at Radio 1/1Xtra I am looking forward to focusing on other projects and new challenges."
The last hour of Jamil's show will be presented in-vision on the Radio 1 website, with the top ten videos being screened in full.
Jamil said: "I am absolutely delighted and honoured. I'm fully aware that I have big shoes to fill and am so excited to get started. I've grown up listening to the official chart so I cannot thank Radio 1 enough for the amazing opportunity."
Edmondson said: "I am delighted that from January I'll be spending my weekends sandwiched between Gemma Cairney and Huw Stephens - it is literally a dream come true, although of course in my actual dream everyone was wearing fewer clothes."
YouTube duo Dan (Danisnotonfire) and Phil (AmazingPhil) take over the new fully-visualised Sunday night Radio 1 request show from 7-9pm.
Dan and Phil said: "We are extremely excited to start this new show! We think it's really changing the idea of a request show and bringing it into the internet generation.
"It's going to be an explosive two hours of great music, lols and interactive content that's completely visualised on the Radio 1 website and we want it to stand out as one of the highlights of the week!"
Comedian Tom Deacon rounds out the new schedule with a new comedy show on Wednesday evenings at 9pm.
"I'm really excited about this new show. I've been doing stand-up comedy for six years and presenting on Radio 1 for the last three, so I'm delighted to be bringing these things together," he said.
"Expect awesome comedians live in the studio, great music, and lots of laughs... I can't wait to get started."
Controller of BBC Radio 1 and BBC Radio 1Xtra Ben Cooper said: "I'd like to thank Vernon and Reggie who have both been excellent ambassadors for Radio 1 and Radio 1Xtra during their ten years with us. I wish them all the very best.
"I'm excited by our brand new shows, which will appeal to the smartphone generation with our audience able to listen, watch and share Radio 1's great content."
The departures follow the exit of breakfast show host Chris Moyles earlier this year. Moyles was replaced by Nick Grimshaw.
All the schedule changes come into effect from the beginning of next year.